Question: The area of trapezoid $ABCD$ is $164 \text{cm}^2$.  The altitude is $8 \text{cm}$, $AB$ is $10 \text{cm}$, and $CD$ is $17 \text{cm}$.  What is $BC$, in centimeters? [asy]
/* AMC8 2003 #21 Problem */
size(2inch,1inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)--(12,0));
label("$A$", (0,0), SW);
label("$D$", (31,0), SE);
label("$B$", (6,8), NW);
label("$C$", (16,8), NE);
label("10", (3,5), W);
label("8", (11,4), E);
label("17", (22.5,5), E);
[/asy]
Solution: Label the feet of the altitudes from $B$ and $C$ as $E$ and $F$ respectively.  Considering right triangles $AEB$ and $DFC$, $AE = \sqrt{10^2 - 8^2} = \sqrt{36} = 6\text{ cm}$, and $FD =
\sqrt{17^2-8^2} = \sqrt{225} = 15\text{ cm}$.   So the area of $\triangle AEB$ is $\frac{1}{2}(6)(8) = 24 \text{ cm}^2$, and the area  of $\triangle DFC$ is $\left(\frac{1}{2}\right) (15)(8) = 60 \text{ cm}^2$. Rectangle $BCFE$ has area $164 - (24 + 60) = 80 \text{ cm}^2$. Because $BE = CF = 8$ cm,  it follows that $BC = \boxed{10\text{ cm}}$. [asy]
/* AMC8 2003 #21 Solution */
size(2inch,1inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((6,8)--(6,0), red+linetype("8 4"));
draw((16,8)--(16,0), red+linetype("8 4"));
label("$A$", (0,0), SW);
label("$D$", (31,0), SE);
label("$B$", (6,8), NW);
label("$C$", (16,8), NE);
label("$E$", (6,0), S);
label("$F$", (16,0), S);
label("10", (3,5), W);
label("8", (6,4), E, red);
label("8", (16,4), E, red);
label("17", (22.5,5), E);
[/asy]